Doxorubicin is a commonly used chemotherapy agent in the treatment of malignant mesothelioma. Onconase is an experimental (not approved by the FDA) drug that has shown activity against malignant mesothelioma. The purpose of this study is to determine whether Onconase, when administered into the blood stream, is better than injection of Doxorubicin into the blood stream for the treatment of malignant mesothelioma. Approximately 160 patients across the country will be enrolled on this study. These patients will receive either Onconase by intravenous infusion over 30 minutes weekly, or Doxorubicin every 3 weeks for a maximum of 6 doses. Treatment will continue for at least 18 weeks or until the tumor is no longer responding to treatment. After treatment has been stopped, patients will be monitored by a physician to determine late side effects and future therapy.